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19521968 Revista de la Unión Matemática Argentina y de la Asociación Física Argentina
19441951 Revista de la Unión Matemática Argentina; órgano de la Asociación Física Argentina
19361944

Volume 67, number 1 (2024)
June 2024
Front matter

Threedimensional $C_{12}$manifolds.
Gherici Beldjilali
The present paper is devoted to threedimensional $C_{12}$manifolds (defined by D. Chinea
and C. Gonzalez), which are never normal. We study their fundamental properties and give
concrete examples. As an application, we study such structures on threedimensional Lie
groups.

1–14 
On an extension of the Newton polygon test for polynomial reducibility.
Brahim Boudine
Let $R$ be a commutative local principal ideal ring which is not integral, $f$ a
polynomial in $R[x]$ such that $f(0) \neq 0$ and $N(f)$ its Newton polygon. If $N(f)$
contains $r$ sides of different slopes, we show that $f$ has at least $r$ different pure
factors in $R[x]$. This generalizes the Newton polygon method over a ring which is not
integral.

15–25 
Summing the largest prime factor over integer sequences.
JeanMarie De Koninck and Rafael Jakimczuk
Given an integer $n\ge 2$, let $P(n)$ stand for its largest prime factor. We examine the
behaviour of $\sum\limits_{n\le x \atop n\in A} P(n)$ in the case of two sets $A$, namely
the set of $r$free numbers and the set of $h$full numbers.

27–35 
New classes of statistical manifolds with a complex structure.
Mirjana Milijević
We define new classes of statistical manifolds with a complex structure. Motivation for
our work is the classification of almost Hermitian manifolds with respect to the covariant
derivative of the almost complex structure, obtained by Gray and Hervella in 1980. Instead
of the LeviCivita connection, we use a statistical one and obtain eight classes of Kähler
manifolds with the statistical connection. Besides, we give some properties of tensors
constructed from covariant derivative of the almost complex structure with respect to the
statistical connection. From the obtained properties, further investigation of statistical
manifolds is possible.

37–45 
Coordinate rings of some $\mathrm{SL}_2$character varieties.
Vicente Muñoz and Jesús Martín Ovejero
We determine generators of the coordinate ring of $\mathrm{SL}_2$character varieties. In
the case of the free group $F_3$ we obtain an explicit equation of the
$\mathrm{SL}_2$character variety. For free groups $F_k$, we find transcendental
generators. Finally, for the case of the $2$torus, we get an explicit equation of the
$\mathrm{SL}_2$character variety and use the description to compute their
$E$polynomials.

47–64 
Spectrality of planar Moran–Sierpinskitype measures.
Qian Li and MinMin Zhang
Let $\{M_n\}_{n=1}^{\infty}$ be a sequence of expanding positive integral matrices with
$M_n= \begin{pmatrix} p_n & 0\\0 & q_n \end{pmatrix}$ for each $n\ge 1$, and let
$D=\left\{\begin{pmatrix} 0\ 0 \end{pmatrix}, \begin{pmatrix} 1\ 0 \end{pmatrix},
\begin{pmatrix} 0\ 1 \end{pmatrix} \right\}$ be a finite digit set in $\mathbb{Z}^2$. The
associated Borel probability measure obtained by an infinite convolution of atomic
measures \[
\mu_{\{M_n\},D}=\delta_{M_1^{1}D}*\delta_{(M_2M_1)^{1}D}*\cdots*\delta_{(M_n\cdots
M_2M_1)^{1}D}*\cdots \] is called a Moran–Sierpinskitype measure. We prove that, under
certain conditions, $\mu_{\{M_n\}, D}$ is a spectral measure if and only if $3\mid p_n$
and $3\mid q_n$ for each $n\geq2$.

65–80 
Using digraphs to compute determinant, permanent, and Drazin inverse of circulant matrices with two parameters.
Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Denis E. Videla
This work presents closed formulas for the determinant, permanent, inverse, and Drazin
inverse of circulant matrices with two nonzero coefficients.

81–106 
Existence and multiplicity of solutions for $p$Kirchhofftype Neumann problems.
Qin Jiang, Sheng Ma, and Daniel Paşca
We establish, based on variational methods, existence theorems for a $p$Kirchhofftype
Neumann problem under the Landesman–Lazer type condition and under the local coercive
condition. In addition, multiple solutions for a $p$Kirchhofftype Neumann problem are
established using a known threecriticalpoint theorem proposed by H. Brezis and L.
Nirenberg.

107–121 
Poincaré duality for Hopf algebroids.
Sophie Chemla
We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As
an application, we recover the Hochschild twisted Poincaré duality of van den Bergh. We
also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson
manifolds by Chen et al.

123–136 
Gorenstein properties of splitbynilpotent extension algebras.
Pamela Suarez
Let $A$ be a finitedimensional $k$algebra over an algebraically closed field $k$. In
this note, we study the Gorenstein homological properties of a splitbynilpotent
extension algebra. Let $R$ be a splitbynilpotent extension of $A$. We provide sufficient
conditions to ensure when a Gorensteinprojective module over $A$ induces a similar
structure over $R$. We also study when a Gorensteinprojective $R$module induces a
Gorensteinprojective $A$module. Moreover, we study the relationship between the
Gorensteinness of $A$ and $R$.

137–144 
Distance Laplacian eigenvalues of graphs, and chromatic and independence number.
Shariefuddin Pirzada and Saleem Khan
Given an interval $I$, let $m_{D^{L} (G)} I$ (or simply $m_{D^{L}} I$) be the number of
distance Laplacian eigenvalues of a graph $G$ which lie
in $I$. For a prescribed interval $I$, we give the bounds for
$m_{D^{L}}I$ in terms of the independence number $\alpha(G)$,
the chromatic number $\chi$, the number of pendant vertices $p$, the number of components
in the complement graph $C_{\overline{G}}$ and the diameter $d$ of $G$. In particular, we
prove that $m_{D^{L}(G) }[n,n+2)\leq \chi1$, $m_{D^{L}(G)}[n,n+\alpha(G))\leq
n\alpha(G)$, $m_{D^{L}(G) }[n,n+p)\leq np$ and discuss the cases where the bounds are
best possible. In addition, we characterize graphs of diameter $d\leq 2$ which satisfy
$m_{D^{L}(G) } (2n1,2n )= \alpha(G)1=\frac{n}{2}1$. We also propose some problems of
interest.

145–159 
Limit behaviors for a $\beta$mixing sequence in the St. Petersburg game.
Yu Miao, Qing Yin, and Zhen Wang
We consider a sequence of nonnegative $\beta$mixing random variables $\{X,X_n :
n\geq1\}$ from the classical St. Petersburg game. The accumulated gains
$S_n=X_1+X_2+\cdots+X_n$ in the St. Petersburg game are studied, and the large deviations
and the weak law of large numbers of $S_n$ are obtained.

161–171 
Trivial extensions of monomial algebras.
María Andrea Gatica, María Valeria Hernández, and María Inés Platzeck
We describe the ideal of relations for the trivial
extension $T(\Lambda)$ of a finitedimensional monomial
algebra $\Lambda$. When $\Lambda$ is, moreover, a
gentle algebra, we solve the converse problem: given an
algebra $B$, determine whether $B$ is the trivial
extension of a gentle algebra. We characterize such
algebras $B$ through properties of the cycles of their
quiver, and show how to obtain all gentle
algebras $\Lambda$ such that $T(\Lambda) \cong B$. We
prove that indecomposable trivial extensions of gentle
algebras coincide with Brauer graph algebras with
multiplicity one in all vertices in the associated Brauer
graph, result proven by S. Schroll.

173–196 
On the Eneström–Kakeya theorem and its various forms in the quaternionic setting.
Abdullah Mir
We study the extensions of the classical Eneström–Kakeya theorem and its various
generalizations regarding the distribution of zeros of polynomials from the complex to the
quaternionic setting. We aim to build upon the previous work by various authors and derive
zerofree regions of some special regular functions of a quaternionic variable with
restricted coefficients, namely quaternionic coefficients whose real and imaginary
components or moduli of the coefficients satisfy suitable inequalities. The obtained
results for this subclass of polynomials and slice regular functions produce
generalizations of a number of results known in the literature on this subject.

197–211 
On the planarity, genus, and crosscap of the weakly zerodivisor graph of commutative rings.
Nadeem ur Rehman, Mohd Nazim, and Shabir Ahmad Mir
Let $R$ be a commutative ring and $Z(R)$ its zerodivisors
set. The weakly zerodivisor graph of $R$, denoted by
$W\Gamma(R)$, is an undirected graph with the nonzero
zerodivisors $Z(R)^*$ as vertex set and two distinct
vertices $x$ and $y$ are adjacent if and only if there
exist $a \in \mathrm{Ann}(x)$ and $b \in \mathrm{Ann}(y)$
such that $ab = 0$. In this paper, we characterize finite
rings $R$ for which the weakly zerodivisor graph
$W\Gamma(R)$ belongs to some wellknown families of graphs.
Further, we classify the finite rings $R$ for which
$W\Gamma(R)$ is planar, toroidal or double toroidal.
Finally, we classify the finite rings $R$ for which
the graph $W\Gamma(R)$ has crosscap at most two.

213–227 
Modular automata.
Thomas N. Hibbard, Camilo A. Jadur, and Jorge F. Yazlle
Let $M$ and $b$ be integers greater than 1, and let
$\mathbf{p}$ be a positive probability vector for the alphabet
$\mathcal{A}_{b}=\{0,\ldots,b1\}$. Let us consider a random
sequence $w_0,w_1,\ldots,w_j$ over $\mathcal{A}_{b}$, where the
$w_i$'s are independent and identically distributed according
to $\mathbf{p}$. Such a sequence represents, in
base $b$, the number $n=\sum_{i=0}^j w_i b^{ji}$. In this
paper, we explore the asymptotic distribution of $n\mbox{ mod
}M$, the remainder of $n$ divided by $M$. In particular,
by using the theory of Markov chains, we show that if $M$ and
$b$ are coprime, then $n\mbox{ mod } M$ exhibits an asymptotic
discrete uniform distribution, independent
of $\mathbf{p}$; on the other hand, when $M$ and $b$ are
not coprime, $n\mbox{ mod }M$ does not necessarily have a
uniform distribution, and we obtain an explicit expression for
this limiting distribution.

229–244 
The principal small intersection graph of a commutative ring.
Soheila Khojasteh
Let $R$ be a commutative ring with nonzero identity. The small intersection graph of $R$,
denoted by $G(R)$, is a graph with the vertex set $V(G(R))$, where $V(G(R))$ is the set of
all proper nonsmall ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if
and only if $I \cap J$ is not small in $R$. In this paper, we introduce a certain subgraph
$PG(R)$ of $G(R)$, called the principal small intersection graph of $R$. It is the
subgraph of $G(R)$ induced by the set of all proper principal nonsmall ideals of $R$. We
study the diameter, the girth, the clique number, the independence number and the
domination number of $PG(R)$. Moreover, we present some results on the complement of the
principal small intersection graph.

245–256 
Sequential optimality conditions for optimization problems with additional abstract set constraints.
Nadia Soledad Fazzio, María Daniela Sánchez, and María Laura Schuverdt
The positive approximate Karush–Kuhn–Tucker sequential condition and the strict constraint
qualification associated with this condition for general scalar problems with equality and
inequality constraints have recently been introduced. In this paper, we extend them to
optimization problems with additional abstract set constraints. We also present an
extension of the approximate Karush–Kuhn–Tucker sequential condition and its related
strict constraint qualification. Furthermore, we explore the relations between the new
constraint qualification and other constraint qualifications known in the literature as
Abadie, quasinormality and the approximate Karush–Kuhn–Tucker regularity constraint
qualification. Finally, we introduce an augmented Lagrangian method for solving the
optimization problem with abstract set constraints and we show that it is possible to
obtain global convergence under the new condition.

257–279 
Properties of the convolution operation in the complexity space and its dual.
José M. HernándezMorales, Netzahualcóyotl C. CastañedaRoldán, and Luz C. ÁlvarezMarín
We give the basic properties of discrete convolution in the space of complexity functions
and its dual space. Two inequalities are identified, and defined in the general context of
an arbitrary binary operation in any weighted quasimetric space. In that setting, some
quasimetric and convergence consequences of those inequalities are proven. Using
convolution, we show a method for building improver functionals in the complexity space.
We also consider convolution in three topologies within the dual space, obtaining two
topological monoids.

281–299 
Primitive decompositions of Dolbeault harmonic forms on compact almostKähler manifolds.
Andrea Cattaneo, Nicoletta Tardini, and Adriano Tomassini
Let $(X,J,g,\omega)$ be a compact $2n$dimensional almostKähler manifold. We prove
primitive decompositions of $\partial$, $\bar\partial$harmonic forms on $X$ in bidegree
$(1,1)$ and $(n1,n1)$ (such bidegrees appear to be optimal). We provide examples showing
that in bidegree $(1,1)$ the $\partial$ and $\bar\partial$decompositions differ.

301–316 
Cofinite modules and cofiniteness of local cohomology modules.
Alireza Vahidi, Ahmad Khaksari, and Mohammad Shirazipour
Let $n$ be a nonnegative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an
ideal of $R$, $M$ a finitely generated $R$module, and $X$ an arbitrary $R$module.
In this paper, we first prove that if $\dim_R(M)\leq{n+2}$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an $(\operatorname{FD}_{ <
n},\mathfrak{a})$cofinite $R$module and
$\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M)):\dim(R/\mathfrak{p})\geq{n}\}$
is a finite set for all $i$. As a consequence, it follows that
$\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M))$ is a finite set for all $i$
when $R$ is a semilocal ring and $\dim_R(M)\leq{3}$. Then, we show that if
$\dim(R/\mathfrak{a})\leq{n+1}$, then $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an
$\operatorname{FD}_{ < n}$ $R$module for all $i$ whenever
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{ < n}$
$R$module for all $i\leq{\dim_R(X)n}$. Finally, in the case that
$\dim(R/\mathfrak{a})\leq{2}$, $X$ is $\mathfrak{a}$torsion, and $n>0$ or
$\operatorname{Supp}_R(X)\cap\operatorname{Var}(\mathfrak{a})\cap\operatorname{Max}(R)$ is
finite, we prove that $X$ is an $(\operatorname{FD}_{ < n},\mathfrak{a})$cofinite
$R$module when $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{
< n}$ $R$module for all $i\leq{2n}$. We conclude with some ordinary
$\mathfrak{a}$cofiniteness results for local cohomology modules
$\operatorname{H}^{i}_{\mathfrak{a}}(X)$.

317–325 
Lorentz ${C}_{12}$manifolds.
Adel Delloum and Gherici Beldjilali
The object of the present paper is to study $C_{12}$structures on a manifold with
Lorentzian metric. We focus here on Lorentzian $C_{12}$structures, emphasizing their
relationship and analogies with respect to the Riemannian case. Several interesting
results are obtained. Next, we study Ricci solitons in Lorentzian $C_{12}$manifolds.

327–337 
Two constructions of bialgebroids and their relations.
Yudai Otsuto
We generalize the construction of face algebras by Hayashi and obtain a left bialgebroid
$\mathfrak{A}(w)$. There are some relations between the left bialgebroid $\mathfrak{A}(w)$
and the generalized Shibukawa–Takeuchi left bialgebroid $A_{\sigma}$.

339–395 
